Physics 41 HW Set 2 Chapter 14 P4 In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x = (5.00 cm) cos(2t + /6) where x is in centimeters and t is in seconds. At t = 0, find (a) the position of the piston, (b) its velocity, and (c) its acceleration. (d) Find the period and amplitude of the motion. (a) xt5.00 cm cos 2 At t 0, x 5.00 cm cos 4.33 cm 6 6 dx (b) vt 10.0 cm s sin 2 At , v5.00 cm s dt 6 dv 2 2 (c) at 20.0 cm s cos 2 At , a17.3 cm s dt 6 22 (d) A 5.00 cm and T 3.14 s 2 P 5 The position of a particle is given by the expression x = (4.00 m) cos(3.00 t + ), where x is in meters and t is in seconds. Determine (a) the frequency and period of the motion, (b) the amplitude of the motion, (c) the phase constant, and (d) the position of the particle at t = 0.250 s. xt4.00 m cos 3.00 Compare this with x Acos t to find (a) 2 f 3.00 1 or f 1.50 Hz T 0.667 s f (b) A 4.00 m (c) rad (d) xt0.250 s 4.00 m cos 1.75 2.83 m P8. A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find (a) the period of its motion, (b) the frequency in hertz, and (c) the angular frequency in radians per second. 12.0 s (a) T 2.40 s 5 11 (b) f 0.417 Hz T 2.40 (c) 2 f 2 0.417 2.62 rad s P18. A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 2.00 J, find (a) the force constant of the spring and (b) the amplitude of the motion. 22 m 200 g , T 0.250 s, E 2.00 J; 25.1 rad s T 0.250 2 (a) km2 0.200 kg 25.1 rad s 126 N m kA2 2 E 2 2.00 (b) EA 0.178 m 2 k 126 P20. A 2.00-kg object is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest with an initial position of xi = 0.200 m, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the object. Where does this maximum speed occur? (d) Find the maximum acceleration of the object. Where does it occur? (e) Find the total energy of the oscillating system. Find (f) the speed and (g) the acceleration of the object when its position is equal to one third of the maximum value. F 20.0 N (a) k 100 N m x 0.200 m k (b) 50.0 rad s so f1.13 Hz m 2 (c) vAm ax 50.00.200 1.41 m s at x 0 22 (d) aAm ax 50.00.200 10.0 m s at xA 11 2 (e) E kA 2 100 0.200 2.00 J 22 8 2 (f) v A22 x 50.0 0.200 1.33 m s 9 220.200 (g) ax 50.0 3.33 m s 3 P29 A physical pendulum in the form of a planar body moves in simple harmonic motion with a frequency of 0.450 Hz. If the pendulum has a mass of 2.20 kg and the pivot is located 0.350 m from the center of mass, determine the moment of inertia of the pendulum about the pivot point. f 0.450 Hz, d 0.350 m , and m 2.20 kg 1 FIG. P15.35 T ; f 2 II2 4 TT2; m gd m gd 2 m gd1 m gd 2.20 9.80 0.350 IT220.944 kg m 2 2 2 44f 4 21 0.450 s P31 A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.0 and then released. What are (a) the maximum speed, (b) the maximum angular acceleration, and (c) the maximum restoring force? What If? Solve this problem by using the simple harmonic motion model for the motion of the pendulum, and then solve the problem more precisely by using more general principles. Using the simple harmonic motion model: Ar 1 m 15 0.262 m 180 g 9.8 m s2 3.13 rad s L 1 m FIG. P15.31 (a) vAm ax 0.262 m 3.13 s 0.820 m s 222 (b) aAm ax 0.262 m 3.13 s 2.57 m s a 2.57 m s2 ar tan 2.57 rad s2 tan r 1 m (c) F m a 0.25 kg 2.57 m s2 0.641 N More precisely, 1 (a) m gh m v2 and hL1 cos 2 vm ax 2 gL 1 cos 0.817 m s (b) I m gLsin m gLsin g sin 2.54 rad s2 m ax mL2 L i (c) Fm ax m gsini 0.250 9.80 sin15.0 0.634 N 34. A very light rigid rod with a length of 0.500 m extends straight out from one end of a meter stick. The stick is suspended from a pivot at the far end of the rod and is set into oscillation. (a) Determine the period of oscillation. Suggestion: Use the parallel-axis theorem from Section 10.5. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long? 43. A 2.00-kg object attached to a spring moves without friction and is driven by an external force F = (3.00 N) sin(2 t). If the force constant of the spring is 20.0 N/m, determine (a) the period and (b) the amplitude of the motion. 39. An 10.6-kg object oscillates at the end of a vertical spring which has a spring constant of 2.05 104 N/m. The effect of air resistance is represented by the damping coefficient b = 3.00 N·s/m. (a) Calculate the frequency of the damped oscillation. (b) By what percentage does the amplitude of the oscillation decrease in each cycle? (c) Find the time interval that elapses while the energy of the system drops to 5.00% of its initial value. P 51A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P15.51). (a) Determine the tensions in the rod at the pivot and at the point P when the system is stationary. (b) Calculate the period of oscillation for small displacements from equilibrium, and determine this period for L = 2.00 m. (Suggestions: Model the object at the end of the rod as a particle and use Eq. 15.28.) Let F represent the tension in the rod. pivot (a) At the pivot, F M g M g 2 M g P L y A fraction of the rod’s weight Mg as well as L y the weight of the ball pulls down on point P. Thus, the tension in the rod at point P is M yy F M g M g M g 1 . LL FIG. P15.51 14 (b) Relative to the pivot, IIIMLMLML 2 2 2 rod ball 33 I For the physical pendulum, T 2 where mM 2 and d is the distance from the pivot m gd to the center of mass of the rod and ball combination. Therefore, L 4 2 MML 2 3L ML 42 L d and T 2 3 . MM 4 3L 3 g 2Mg 4 4 2 2.00 m For L 2.00 m , T 2.68 s . 3 9.80 m s2 Discussion Problems: 32, 57, 59, 67, 74 32. Consider the physical pendulum of Figure 15.18. (a) If its moment of inertia about an axis passing through its center of mass and parallel to the axis passing through its pivot point is I md 2 T 2 CM mgd ICM, show that its period is where d is the distance between the pivot point 2 and center of mass. (b) Show that the period has a minimum value when d satisfies md = ICM. 15.57 A horizontal plank of mass m and length L is pivoted at one end. The plank’s other end is supported by a spring of force constant k. The plank is displaced by a small angle from its horizontal equilibrium position and released. (a) Show that it moves with simple harmonic motion with an angular frequency = 3k/m . (b) Evaluate the frequency if the mass is 5.00 kg and the spring has a force constant of 100 N/m. P15.57 (a) The problem tells us that the plank and spring are at equilibrium when the plank is horizontal.